Digital Pulse Amplitude Modulation (PAM)

Outline Introduction Pulse shaping Pulse shaping filter bank Design tradeoffs FIR 0 FIR 1 FIR L-1 Symbol recovery

4-level PAM Constellation Map Introduction 4-level PAM Constellation Map d d 3 d 3 d 00 01 10 11 symbol of bits symbol ampl. Convert bit stream into pulse stream Group stream of bits into symbols of J bits Represent symbol of bits by unique amplitude Scale pulse shape by that amplitude M-level PAM or simply M-PAM (M = 2J) Symbol period is Tsym and bit rate is J fsym Impulse train has impulses separated by Tsym Pulse shape may last one or more symbol periods Serial/ Parallel Map to PAM constellation an 1 J bit stream J bits per symbol Pulse shaper gTsym(t) s*(t) symbol amplitude baseband waveform Impulse modulator impulse train d is a voltage

Pulse Shaping Without pulse shaping One impulse per symbol period Infinite bandwidth used (not practical) k is a symbol index Limit bandwidth by pulse shaping (FIR filtering) Convolution of discrete-time signal ak and continuous-time pulse shape For pulse shape lasting Ng Tsym seconds, Ng pulses overlap in each symbol period See lecture slide 7-5 Serial/ Parallel Map to PAM constellation an 1 J bit stream J bits per symbol Pulse shaper gTsym(t) s*(t) symbol amplitude baseband waveform Impulse modulator impulse train

2-PAM Transmission 2-PAM example (right) Highest frequency ½ fsym Truncated raised cosine pulse with peak value of 1 What are d and Tsym ? How does maximum amplitude relate to d? Highest frequency ½ fsym Alternating symbol amplitudes +d, -d, +d, … Serial/ Parallel Map to PAM constellation an 1 J bit stream J bits per symbol Pulse shaper gTsym(t) s*(t) symbol amplitude baseband waveform Impulse modulator impulse train

Baseband PAM Transmission Code % Discrete-time baseband PAM signal mmax = N*L; v = zeros(1,mmax); v(1:L:end) = symAmp; % interpolation s = conv(v, g); % pulse shaping slength = length(s); % trim result s = s(midpt+1:slength-midpt+1); % Interpretation in continuous time Tsym = 4; % Symbol period in sec fsym = 1/Tsym; % Symbol rate in Hz fs = L*fsym; % Sampling rate in Hz Ts = 1/fs; % Sampling time in sec % Plots Mmax = length(s); m = 0 : (Mmax-1); t = m*Ts; Nmax = Mmax / L; n = 0 : (Nmax-1); figure; plot(t,s); hold on; stem(n*Tsym,symAmp); hold off; xlim( [0 (Nmax-(Ng/2))*Tsym-Ts] ); ymax = Ng * (M-1) * d / 2; ylim( [-ymax ymax] ); xlabel('Time (ms)'); title('Baseband PAM Signal s(t)'); % Baseband PAM Signal Generation % by Prof. Brian L. Evans % The University of Texas at Austin % Spring 2018 % % m is sample index % n is symbol index % Simulation parameters N = 25; % Number symbol periods % Pulse shape g[m] Ng = 4; % Number symbol periods L = 20; % Samples/symbol period f0 = 1/L; midpt = Ng*L/2; m = (-midpt) : (midpt-1); g = sinc(f0*m); % Adjust for group delay N = N + (Ng/2); % M-level PAM symbol amplitudes d = 1; M = 2; ioffset = M + 1; symAmp = (2*randi(M,[1,N]) - ioffset)*d;

4-PAM Transmission 4-PAM example (right) Highest frequency ½ fsym Truncated raised cosine pulse with peak value of 1 What are d and Tsym ? How does maximum amplitude relate to d? Highest frequency ½ fsym Alternating symbol amplitudes +d, -d, +d, ... or +3d, -3d, +3d, ... Serial/ Parallel Map to PAM constellation an 1 J bit stream J bits per symbol Pulse shaper gTsym(t) s*(t) symbol amplitude baseband waveform Impulse modulator impulse train

PAM Transmission Transmitted signal Sample at sampling time Ts : let t = (n L + m) Ts L samples per symbol period Tsym i.e. Tsym = L Ts n is the index of the current symbol period being transmitted m is a sample index within nth symbol (i.e., m = 0, 1, …, L-1) Pulse shaper gTsym[m] Serial/ Parallel Map to PAM constellation L D/A 1 J an s*(t) bit stream J bits per symbol symbol amplitude impulse train baseband waveform baseband waveform

Pulse Shaping Block Diagram Upsampling by L denoted as L Outputs input sample followed by L-1 zeros Upsampling by L converts symbol rate to sampling rate Pulse shaping (FIR) filter gTsym[m] Fills in zero values generated by upsampler Multiplies by zero most of time (L-1 out of every L times) an s*(t) L gTsym[m] D/A Transmit Filter symbol rate sampling rate sampling rate cont. time cont. time

Digital Interpolation Example Upsampling by 4 (denoted by L) Output input sample followed by 3 zeros Four times the samples on output as input Increases sampling rate by factor of 4 FIR interpolation filter (lowpass) Prior to upsampling, fmax is 22.05 kHz At filter input, wmax = 2 p fmax / fsampling = p / 4 = p / L Filter specifications: wstop < p / 4 and wpass = 0.9 wstop Digital 4x Oversampling Filter 16 bits 44.1 kHz 28 bits 176.4 kHz 4 FIR Filter 16 bits 176.4 kHz 1 2 Input to Upsampler by 4 n n’ Output of Upsampler by 4 1 2 3 4 5 6 7 8 1 2 Output of FIR Filter 3 4 5 6 7 8 n’

Pulse Shaping Filter Bank Example L = 4 samples per symbol Pulse shape g[m] lasts for 2 symbols (8 samples) bits …a2a1a0 …000a1000a0 encoding ↑4 g[m] x[m] s[m] s[m] = x[m] * g[m] s[0] = a0 g[0] s[1] = a0 g[1] s[2] = a0 g[2] s[3] = a0 g[3] s[4] = a0 g[4] + a1 g[0] s[5] = a0 g[5] + a1 g[1] s[6] = a0 g[6] + a1 g[2] s[7] = a0 g[7] + a1 g[3] L polyphase filters operating at symbol rate {g[0],g[4]} {g[1],g[5]} {g[2],g[6]} {g[3],g[7]} s[m] …,s[4],s[0] …,s[5],s[1] …,s[6],s[2] …,s[7],s[3] …,a1,a0 m=0 Commutator (Periodic) Filter Bank

Pulse Shaping Filter Bank Simplify by avoiding multiplication by zero Split long pulse shaping filter into L short polyphase filters operating at symbol rate an L gTsym[m] D/A Transmit Filter symbol rate sampling rate sampling rate cont. time cont. time gTsym,0[n] s(Ln) gTsym,1[n] D/A Transmit Filter an s(Ln+1) Filter Bank Implementation gTsym,L-1[n] s(Ln+(L-1))

Pulse Shaping Filter Bank Example Pulse length 24 samples and L = 4 samples/symbol Derivation: let t = (n + m/L) Tsym Define mth polyphase filter Four six-tap polyphase filters (next slide) Six pulses contribute to each output sample

Pulse Shaping Filter Bank Example 24 samples in pulse gTsym,0[n] 4 samples per symbol Polyphase filter 0 impulse response is first sample of the pulse shape and every fourth sample after that Polyphase filter 0 has only one non-zero sample.

Pulse Shaping Filter Bank Example 24 samples in pulse gTsym,1[n] 4 samples per symbol Polyphase filter 1 impulse response is second sample of the pulse shape and every fourth sample after that

Pulse Shaping Filter Bank Example 24 samples in pulse gTsym,2[n] 4 samples per symbol Polyphase filter 2 impulse response is third sample of the pulse shape and every fourth sample after that

Pulse Shaping Filter Bank Example 24 samples in pulse gTsym,3[n] 4 samples per symbol Polyphase filter 3 impulse response is fourth sample of the pulse shape and every fourth sample after that

Pulse Shaping Design Tradeoffs Computation in MACs/s Memory size in words Memory reads in words/s Memory writes in words/s Direct structure (slide 13-9) (L Ng)(L fsym) Filter bank structure (slide 13-12) L Ng fsym fsym symbol rate L samples/symbol Ng duration of pulse shape in symbol periods

Pulse Shaping Design Tradeoffs FIR filter with N coefficients to compute output Read current input value x[n] Read pointer to newest input sample in circular buffer Read previous N-1 input values Read N filter coefficients h Compute the output value y[n] Write current input value to circular buffer to oldest sample Write (update) the newest pointer in circular buffer Write output value N multiplications-additions, 2N + 1 reads, 3 writes

Pulse Shaping Design Tradeoffs Direct structure (slide 13-9) FIR filter has L Ng coefficients (N = L Ng) FIR filter: N multiplications-additions, 2N + 1 reads, 3 writes FIR filter runs at sampling rate L fsym Upsampler reads at symbol rate and writes at sampling rate Filter bank structure (slide 13-12) L polyphase FIR filters with Ng coefficients each FIR filter: Ng multiplications-additions, 2Ng + 1 reads, 3 writes Each polyphase FIR filter runs at symbol rate fsym Commutator reads and writes at sampling rate L fsym

Optional Symbol Clock Recovery Transmitter and receiver normally have different oscillator circuits Critical for receiver to sample at correct time instances to have max signal power and min ISI Receiver should try to synchronize with transmitter clock (symbol frequency and phase) First extract clock information from received signal Then either adjust analog-to-digital converter or interpolate Next slides develop adjustment to A/D converter Also, see Handout M in the reader

s*(t) is transmitted signal Periodic with period Tsym Optional Symbol Clock Recovery g1(t) is impulse response of LTI composite channel of pulse shaper, noise-free channel, receive filter s*(t) is transmitted signal g1(t) is deterministic E{ak am} = a2 d[k-m] Periodic with period Tsym Receive B(w) Squarer BPF H(w) PLL x(t) q(t) q2(t) p(t) z(t)

Symbol Clock Recovery Fourier series representation of E{ p(t) } Optional Symbol Clock Recovery Fourier series representation of E{ p(t) } In terms of g1(t) and using Parseval’s relation Fourier series representation of E{ z(t) } where Receive B(w) Squarer BPF H(w) PLL x(t) q(t) q2(t) p(t) z(t)

Symbol Clock Recovery With G1(w) = X(w) B(w) Optional Symbol Clock Recovery With G1(w) = X(w) B(w) Choose B(w) to pass  ½wsym  pk = 0 except k = -1, 0, 1 Choose H(w) to pass wsym  Zk = 0 except k = -1, 1 B(w) is lowpass filter with wpassband = ½ wsym H(w) is bandpass filter with center frequency wsym Receive B(w) Squarer BPF H(w) PLL x(t) q(t) q2(t) p(t) z(t)